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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2026 • Question 5

Question

Math 6310 • Algebra Spring 2026 • Question 5

Let \(M_n(\mathbb {C})\) be the algebra of \(n\) by \(n\) complex matrices. Let \(T \in M_n(\mathbb {C})\) such that \(T^m=1\) for some \(m>0\). Prove that \(T\) is conjugate to a diagonal matrix.

Raw LaTeX

Let \(M_n(\mathbb{C})\) be the algebra of \(n\) by \(n\) complex matrices. Let
\(T \in M_n(\mathbb{C})\) such that \(T^m=1\) for some \(m>0\). Prove that \(T\)
is conjugate to a diagonal matrix.

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization